Spectral contraction from Gaussian beams to Helmholtz waves

Abstract

We introduce the ratio between the transverse Helmholtz length and the paraxial Gaussian-beam width as the physical parameter governing the contraction from separable Gaussian beams to transverse Helmholtz waves. For a fixed transverse Helmholtz shell within the paraxial angular spectrum, the large-Rayleigh-range limit drives our optical scale ratio towards zero, while the higher-modal-order limit preserves the shell, transforming the isotropic and astigmatic paraxial oscillator eigenvalue problems into the transverse Helmholtz unit-shell eigenvalue problem. As our optical scale ratio vanishes, it realizes the algebraic Inönü-Wigner contraction of the isotropic Gaussian-beam $\mathrm{su}(2)$ algebra to the quadratic fixed-shell realization of the transverse Helmholtz $\mathrm{e}(2)$ algebra; conversely, finite nonzero values parametrize the Cayley-Klein deformation. Under the joint limit of vanishing scale ratio and higher modal order, paraxial Hermite-, Laguerre-, Ince-, and Boyer-Wolf-Gaussian beams undergo a spectral contraction in which their two modal numbers fix the shell and determine the single separation parameter labeling the corresponding Cartesian, Bessel, Mathieu, and Weber transverse Helmholtz waves, while preserving parity. Our spectral contraction extends to the wave operators, symmetry algebras, separation operators, separated equations, and separation spectra.

arXiv comment13 pages, 2 figures